The correct form for the Green's function for the vector Helmholtz equation is

(13.17)

(where
is a Green's function for the scalar
IHE, that is:

(13.18)

for
. The identity tensor transforms a
vector on the right into the same vector, so this seems like a trivial
definition. However, the point is that we can now expand the identity tensor
times the scalar Green's function in vector spherical harmonics or Hansen functionsdirectly!

We get:

(13.19)

In all cases the ``*''s are to be considered sliding, able to apply to
the
only of either term under an integral.

I do not intend to prove a key element of this assertion (that the
products of the
involved reduce to Legendre
polynomials in the angle between the arguments times the identity
tensor) in class. Instead, I leave it as an exercise. To get you
started, consider how similar completeness/addition theorems are proven
for the spherical harmonics themselves from the given orthonormality
relation.

With these relations in hand, we end our mathematical digression into
vector spherical harmonics and the Hansen solutions and return to the
land of multipolar radiation.